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arXiv · 2105.05250

Several special cases of a square problem

Yang Ji
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Abstract

Here is a square problem: in a unit square, is there a point with four rational distances to the vertices? A probability argument suggests a negative answer. This paper proves several special cases of the square problem: if the point sits on the diagonals, the midlines or the edges of the square, or the side-length of the square is n times the distance from the point to one side (both $n$ and $(n^2+4)$ are prime numbers), the distances from this point to the four vertices can not be all rational. However, this paper does not prove a more general situation. The proof here can be extended to the whole plane, instead of being limited to the interior of the square. Key Words: discrete Geometry; rational distance; a square problem.

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Yang Ji. 2021-05-13. Several special cases of a square problem. https://arxiv.org/abs/2105.05250

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